Quillen's theorems A and B

In topology, a branch of mathematics, Quillen's Theorem A gives a sufficient condition for the classifying spaces of two categories to be homotopy equivalent. Quillen's Theorem B gives a sufficient condition for a square consisting of classifying spaces of categories to be homotopy Cartesian. The two theorems play central roles in Quillen's Q-construction in algebraic K-theory and are named after Daniel Quillen.

The precise statements of the theorems are as follows.

In general, the homotopy fiber of $$Bf: BC \to BD$$ is not naturally the classifying space of a category: there is no natural category $$Ff$$ such that $$FBf = BFf$$. Theorem B constructs $$Ff$$ in a case when $$f$$ is especially nice.